## Section 5 — Diagnostic Expectations and the Financial Accelerator

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---

### Diagnostic Expectations: A Primer
- Core idea:
  - Diagnostic Expectations (DE) introduced by Bordalo et al. (2018) departs from Rational Expectations (RE) by over-weighting news received in the most recent periods, generating excessively volatile expectations and enhanced shock amplification.
- Intuition:
  - An agent assessing future value of a random variable X conditional on data D selectively recalls past states most similar to D (memory is associative).
  - Over-sampling of representative past states leads to probability overestimation (example: Irish hair color — see Bordalo et al. (2022) and numerical example based on Gennaioli and Shleifer (2010) where probability of X=red is 10 percent overall, 10 percent when D=Irish, and 2 percent when D=Not Irish).
- Formalization (following Bordalo et al. (2018)):
  - AR(1) process: z_t = ρ z_{t-1} + ε_t.
  - Diagnostic distribution:
    f^θ_t(z_{t+1}) = f(z_{t+1}|z_t = \breve z_t) [ f(z_{t+1}|z_t = \breve z_t) / f(z_{t+1}|z_t = ρ \breve z_{t-1}) ]^{θ} C
    - \breve z_t denotes realization of z_t, C is normalization constant, and θ ≥ 0 is the diagnosticity parameter.
    - When θ = 0, RE holds; when θ > 0, RE is distorted by the likelihood ratio governed by θ.
  - Tractable expectation formula:
    E^θ_t[z_{t+1}] = E_t[z_{t+1}] + θ ( E_t[z_{t+1}] − E_{t-1}[z_{t+1}] )
    - Surprise term: ξ_{z_{t+1}} = E_t[z_{t+1}] − E_{t-1}[z_{t+1}].
  - Assumptions and notes:
    - ε_t ~ N(0, σ^2), ρ ∈ [0, 1).
    - Reference event carries no news: z_t = ρ \breve z_{t-1} (ε_t = E[ε_t] = 0).
    - DE operator introduces behavioral inattention: predetermined variables are observed with a lag and cannot be treated as constants.

### Key implications of DE
- DE produces additional “surprise” terms in expectations: E^θ_t[·] = E_t[·] + θ (E_t[·] − E_{t-1}[·]).
- These surprise terms increase the endogenous propagation of shocks and can amplify the financial accelerator mechanism.
- DE generates excessive expectation volatility relative to RE, raising macroeconomic variability and complicating stabilization.

---

### Model Overview and Structure

### High-level framework
- Built on Kiyotaki and Moore (1997), Bernanke et al. (1999), and Iacoviello (2005), augmented by Diagnostic Expectations.
- Economy: discrete time, infinite horizon, four agent types:
  - Patient households
  - Impatient entrepreneurs (lower discount rate)
  - Retailers
  - Central bank
- Entrepreneurs produce a homogeneous good using labor and collateralized real estate, creating the financial accelerator mechanism.

### Patient households (notation retained)
- Choice variables: C^P_t, H^P_t, L^P_t, B^P_t.
- Utility: U^P_t = log(C^P_t) + φ^P_t log(H^P_t) − (L^P_t)^η / η.
- Nominal budget constraint:
  P_t C^P_t + Q_t H^P_t + R_{t-1} B^P_{t-1} ≤ B^P_t + W_t L^P_t + Q_t H^P_{t-1} + P_t F_t − P_t T^P_t
- Consumption Euler under DE:
  1 / (π_t C^P_t) = β E^θ_t[ 1 / C^P_{t+1} · R_t / π_{t+1} · 1 / π_t ]
  - DE operator introduces surprise terms in expected consumption and inflation.
  - Inflation: π_t = P_t / P_{t-1}; under DE, P_t cannot be freely moved into E^θ_t due to lagged reference dependence.
- Housing FOC with DE:
  q_t / C^P_t = φ^P H^P_t + β E^θ_t[ q_{t+1} / C^P_{t+1} ]
  - q_t = Q_t / P_t; expected real house price includes a DE surprise term.

### Entrepreneurs
- Production function:
  Y^w_t = A_t (H^E_{t-1})^ν (L^E_t)^{1−ν}
  - Intermediate output sold at price P^w_t; X_t = P_t / P^w_t is the markup. A_t is TFP.
- Utility: U^E_t = log(C^E_t) with discount factor γ < β.
- Nominal budget constraint:
  P^w_t Y^w_t − W_t L^E_t + B^E_t + Q_t H^E_{t-1} ≤ P_t C^E_t + Q_t H^E_t + R_{t-1} B^E_{t-1}
- Collateral constraint:
  B^E_t ≤ m E^θ_t[ q_{t+1} H^E_t R_t / π_{t+1} / π_t ]
  - m interpreted as loan-to-value (LTV) ratio.
- Euler and pricing under DE:
  - Consumption Euler:
    1 / (π_t C^E_t) = γ E^θ_t[ 1 / C^E_{t+1} · R_t / π_{t+1} · 1 / π_t ] + λ_t q_t / C^E_t
  - Borrowing constraint links house prices and interest rates to maximum borrowing; DE implies surprise terms in expected consumption, inflation, and real house price that amplify endogenous shock propagation.

### Final goods and retailers
- Retailers face Rotemberg (1982) adjustment cost, yielding log-linearized NK Phillips curve with DE:
  ˆπ_t = β E^θ_t[ ˆπ_{t+1} ] − (ε − 1)/ψ ˆx_t + ε^π_t
  - Expected inflation under DE includes a surprise term, increasing endogenous variability.

### Central bank
- Linearized interest rate rule:
  ˆR_t = ρ_R ˆR_{t-1} + (1 − ρ_R) [ ω_π ˆπ_t + ω_Y ˆY_t ] + ε^R_t
- Baseline rule may be modified to incorporate house price dynamics.

---

### Model Dynamics Under Diagnostic Expectations

### Log-linearized consumption relation (patient households)
- Euler (log-linearized):
  E^θ_t[ ˆC^P_{t+1} ] − ˆC^P_t = E^θ_t[ ˆR_t − ˆπ_{t+1} − ˆπ_t ] − ˆπ_t
- DE decomposition:
  - E^θ_t[ ˆC^P_{t+1} ] = E_t[ ˆC^P_{t+1} ] + θ ( E_t[ ˆC^P_{t+1} ] − E_{t-1}[ ˆC^P_{t+1} ] ) with surprise ξ^{C^P}_{t+1} = E_t[ ˆC^P_{t+1} ] − E_{t-1}[ ˆC^P_{t+1} ].
  - E^θ_t[ ˆπ_{t+1} ] = E_t[ ˆπ_{t+1} ] + θ ξ^π_{t+1}, where ξ^π_{t+1} = E_t[ ˆπ_{t+1} ] − E_{t-1}[ ˆπ_{t+1} ].
  - Additional term θ (E_t[ ˆπ_t ] − E_{t-1}[ ˆπ_t ]) = θ ξ^π_t appears from log-linearization.
- Combined representation:
  E^θ_t[ ˆC^P_{t+1} ] − ˆC^P_t = E_t[ ∆ ˆC^P_{t+1} ] + ξ^{C^P}_{t+1}
  = E_t[ ˆRR_{t+1} ] + θ ξ^R_t − θ ξ^π_{t+1} − θ ξ^π_t
  - Where ˆRR_{t+1} = ˆR_t − ˆπ_{t+1}.
- Implications:
  - DE introduces surprise terms in expected consumption growth and real interest rates (ξ terms), generating additional endogenous dynamics.
  - Marginal propensity to consume is higher under DE than RE: shocks boosting current income induce overly optimistic views of future output → overconsumption → subsequent disappointment and stronger contraction once reference expectations are revised.
  - Surprise in expected inflation and current inflation can lower the real interest rate (expansionary channel), interacting with consumption surprises to amplify consumption growth swings.
  - Financial accelerator amplifies these DE-driven swings, producing larger business cycle fluctuations.
  - For supply shocks, DE strengthens the debt deflation channel via surprises in expected and current inflation, which can reduce entrepreneurs’ debt burden and dampen shocks.

---

### Model Parameterization (Section 3.6)

### Calibration approach and key parameters
- Parameterization follows Iacoviello (2005) with exceptions noted in Table 1.
- Pricing uses Rotemberg (1982) rather than Calvo (1983), requiring:
  - Elasticity of substitution for intermediate goods: 11 (steady state markup of 10 percent).
  - Price adjustment parameter: 100.
- Two key DE parameters:
  - J (time lag over which the RE revision is defined): benchmark J = 1.
  - Diagnosticity parameter θ: benchmark θ = 0.75 (in line with Bordalo et al. (2018) and L’Huillier et al. (2023)).
- Benchmark monetary policy rule parameters (reported in text):
  - Weight on inflation ωπ = 2.
  - Weight on output gap ωY = 0.12.
  - Interest rate smoothing ρR = 0.73.
- Shock processes are calibrated and summarized at the end of Table 1.
- Sensitivity tests consider alternative parameterizations affecting DE strength, exogenous variation, and monetary policy rules.

### Table 1 — Selected parameter values (as reported)
- Discount factors:
  - Patient household β 0.99
  - Entrepreneurs γ 0.98
- Weight on housing (services) φ 0.03
- Labor supply aversion η 1.01
- Housing share ν 0.1
- Elasticity of substitution ε 20
- Price adjustment cost ψ 140
- Loan-to-value ratio m 0.89
- Diagnosticity parameter θ 0.75
- Monetary policy rule (Table 1 entries):
  - Weight on inflation ωπ 1.5
  - Weight on output gap ωY 0.12
  - Interest rate smoothing ρR 0.73
- Shocks (standard deviation and persistence):
  - Monetary policy: σR 0.25; ρR 0.25
  - Cost push: σP 0.25; ρP 0.9
  - Total factor productivity: σA 0.25; ρA 0.9
  - Housing preference: σH 0.25; ρH 0.9

### Sensitivity and alternative calibrations
- Baseline DE calibration: θ = 0.75, J = 1.
- Alternatives considered: J ∈ {1, 4} and θ ∈ {1, 2}.
- More distant memory (higher J) and θ > 1 produce more severe recessions.
- Example comparisons (reported):
  - Under baseline: FA results in output decline of ́0.8 percent versus ́0.7 percent under baseline without FA (difference 0.1 percentage points).
  - DE (θ = 2, J = 4) results in output decline of ́2.5 percent (difference of 1.8 percentage points versus baseline).
  - Sum of individual contributions = ́1.9 percentage points; combined DE and FA yields an output decline of ́3.1 percent (difference versus baseline ́2.5 percentage points) — i.e., the whole is greater than the sum of parts.

---

### Main Dynamic Results, Mechanisms, and Quantitative Examples

### Interaction of DE and the Financial Accelerator (FA)
- Combining DE with FA generates mutually reinforcing shock amplification.
- Demand shocks:
  - DE and FA jointly amplify demand shocks via:
    - Asset price channel: declines in house prices tighten collateral constraints.
    - Debt deflation channel: lower inflation raises real debt service costs, depressing entrepreneurial consumption.
    - Overreaction under DE: agents become overly pessimistic after negative surprises, increasing the marginal propensity to consume and deepening recessions.
- Supply (cost-push) shocks:
  - FA tends to dampen supply shocks via the debt deflation channel because higher inflation loosens collateral constraints.
  - Under DE, cost-push shocks produce larger stagflationary outcomes (higher inflation and deeper recession); DE strengthens the debt deflation channel.

### Quantitative examples for a 1% (annualized) monetary policy shock
- Baseline (RE, no FA): immediate output decline of ́0.7 percent; cumulative output decline of ́2 percent after 30 quarters.
- RE with FA: immediate output decline of ́0.8 percent; cumulative decline of ́3.5 percent after 30 quarters.
- DE without FA: immediate output decline of ́1.2 percent (about twice the baseline immediate decline).
- DE with FA: immediate output decline of ́1.3 percent (versus ́0.7 baseline).
- Key qualitative insight: DE strengthens the asset price channel and amplifies surprises feeding into FA dynamics, producing larger declines in house prices and output.

---

### Policy Implications — Simple Monetary Policy Rules (Section 4.3)

### Policy objective and generalized rule
- Objective function:
  W = E0 Σ_{t=0}^∞ β^t (ˆπ_t^2 + Λ_y ˆY_t^2 + Λ_h ˆπ_{t,h}^2)
  - Λ_y is relative weight on output gap; Λ_h is relative weight on housing prices inflation.
- Generalized interest rate rule:
  ˆR_t = ρ_R ˆR_{t-1} + (1-ρ_R) (ω_π ˆπ_t + ω_Y ˆY_{t-1} + ω_q ˆq_t + ω_{∆Q} ˆ∆Q_t) + ε_{R,t}
  - Allows responses to level of house prices (ˆq_t) and growth of house prices (ˆ∆Q_t).

### Rules evaluated
- Five simple rules:
  1. Flexible inflation targeting (FIT): ω_π ≥ 0, ω_Y ≥ 0, ω_q = ω_{∆Q} = 0.
  2. Strict inflation targeting (SIT): ω_π ≥ 0, ω_Y = ω_q = ω_{∆Q} = 0.
  3. Price-level targeting (PLT): target ˆP_t instead of ˆπ_t.
  4. Targeting level of house prices: ω_q ≥ 0 (with ω_π ≥ 0, ω_Y ≥ 0).
  5. Targeting house price growth: ω_{∆Q} ≥ 0 (with ω_π ≥ 0, ω_Y ≥ 0).
- Four model variants analyzed: RE and DE, each with and without FA.

### Main findings across models and rules
- Baseline (RE without FA):
  - Optimal simple rule is FIT, coinciding with SIT in this case.
  - PLT gives greater output stabilization but slightly higher inflation volatility, increasing welfare losses relative to FIT.
- Introduction of FA:
  - Welfare losses increase versus baseline.
  - FIT outperforms SIT and PLT when output is considered.
  - Targeting the level of house prices offers no merit: reported weight on level is zero in the row noted (φ_q = 0), making level-targeting equivalent to FIT in those cases.
  - Targeting house price growth yields the lowest welfare losses:
    - Taming asset price channel by targeting house price growth reduces output volatility with only a slight increase in inflation volatility.
    - Advantage: policymaker need not infer asset fundamental values; targeting growth is robust to information asymmetries.
- DE variants:
  - DE without FA: welfare losses associated with policies increase relative to RE; PLT yields most favorable welfare outcome.
  - DE with FA:
    - No value in targeting level of house prices (overlap between column [15] and FIT under column [12]).
    - PLT superior to FIT and SIT.
    - Policy rule targeting house price growth performs best by welfare score:
      - The rule responds aggressively to fluctuations in house price growth (φ_{∆Q} = 2.18), suppressing the asset price channel while accommodating some inflation volatility to harness debt deflation benefits (strengthened under DE).

### Interpretation and recommendations
- Consistent result: no case for targeting the level of house prices within this framework.
- Merits of targeting house price growth:
  - Lowers welfare costs and improves stabilization by reducing output volatility more than it raises inflation volatility.
  - Robust to information frictions because it avoids inferring asset fundamental values.
- FIT (including response to output) generally outperforms strict inflation-only strategies when FA is present.

### Sensitivity exercises and robustness
- Robustness checks (Appendices D and E) confirm main takeaways:
  - Benefits of targeting house price growth become more pronounced when DE amplification is stronger (e.g., J = 4, θ = 2).
  - Introducing indexed debt:
    - Indexed debt can negate wealth effects and impacts on the collateral constraint associated with inflation but does not remove current inflation surprises associated with DE; it has limited effect and does not alter main conclusions.
  - Different shock combinations (only supply shocks or only cost-push shock) do not overturn main findings.
  - Lower loan-to-value ratio of 50 percent (tested): targeting house price growth still yields best welfare outcomes.

### Concluding policy message
- The financial accelerator with Diagnostic Expectations generates mutually reinforcing shock amplification, worsening the inflation-output volatility trade-off.
- Across model variants and robustness checks, targeting house price growth consistently reduces welfare costs and improves stabilization properties.
- Two avenues for future research:
  - Assess whether the case for leaning against the wind by targeting house price growth holds when macroprudential tools exist and can be effectively implemented by a relevant authority.
  - Empirically evaluate a model combining DE, financial frictions, and other empirically-relevant rigidities.

*Section 5 and Sections 3.6 and 4.3, wpiea2024132-print-pdf — The Diagnostic Financial Accelerator — Working Paper No. WP/2024/132.*

### Section 5.

### Section 5.

### Diagnostic Expectations: A Primer
- Diagnostic Expectations (DE) introduced by Bordalo et al. (2018) departs from Rational Expectations (RE) by over-weighting news received in the most recent periods, generating excessively volatile expectations and enhanced shock amplification.
- Intuition:
  - An agent assessing future value of a random variable X conditional on data D selectively recalls past states most similar to D (memory is associative).
  - Over-sampling of representative past states leads to probability overestimation (example: Irish hair color — see Bordalo et al. (2022) and numerical example based on Gennaioli and Shleifer (2010) where probability of X=red is 10 percent overall, 10 percent when D=Irish, and 2 percent when D=Not Irish).
- Formalization (following Bordalo et al. (2018)):
  - Consider AR(1): z_t = ρ z_{t-1} + ε_t.
  - Diagnostic distribution:
    f^θ_t(z_{t+1}) = f(z_{t+1}|z_t = \breve z_t) [ f(z_{t+1}|z_t = \breve z_t) / f(z_{t+1}|z_t = ρ \breve z_{t-1}) ]^{θ} C
    - \breve z_t denotes realization of z_t, C is normalization constant, and θ ≥ 0 is the diagnosticity parameter.
    - When θ = 0, RE holds; when θ > 0, RE is distorted by the likelihood ratio governed by θ.
  - Tractable expectation formula:
    E^θ_t[z_{t+1}] = E_t[z_{t+1}] + θ ( E_t[z_{t+1}] − E_{t-1}[z_{t+1}] )
    - The surprise term defined as ξ_{z_{t+1}} = E_t[z_{t+1}] − E_{t-1}[z_{t+1}] arises because the reference expectation is revised with a one-period lag (J=1 in exposition).
  - Assumptions and notes:
    - ε_t ~ N(0, σ^2), ρ ∈ [0, 1).
    - Reference event carries no news: z_t = ρ \breve z_{t-1} (ε_t = E[ε_t] = 0).
    - The DE operator introduces behavioral inattention: predetermined variables are observed with a lag and cannot be treated as constants.

### Model Overview
- Framework builds on Kiyotaki and Moore (1997), Bernanke et al. (1999), and Iacoviello (2005), augmented by Diagnostic Expectations.
- Economy: discrete time, infinite horizon, four agent types:
  1. Patient households
  2. Impatient entrepreneurs (lower discount rate)
  3. Retailers
  4. Central bank
- Entrepreneurs produce a homogeneous good using labor and collateralized real estate, creating the financial accelerator mechanism.

### Patient Households
- Choice variables: consumption C^P_t, housing H^P_t, labor L^P_t, lending B^P_t.
- Utility: U^P_t = log(C^P_t) + φ^P_t log(H^P_t) − (L^P_t)^η / η.
- Nominal budget constraint:
  P_t C^P_t + Q_t H^P_t + R_{t-1} B^P_{t-1} ≤ B^P_t + W_t L^P_t + Q_t H^P_{t-1} + P_t F_t − P_t T^P_t
  - Definitions: P_t price level, Q_t price of housing, R_t nominal interest rate, W_t wages, F_t profits, T^P_t transfers, φ^P_t housing preference.
- Consumption Euler equation under DE:
  1 / (π_t C^P_t) = β E^θ_t[ 1 / C^P_{t+1} · R_t / π_{t+1} · 1 / π_t ]
  - DE operator E^θ_t[·] introduces surprise terms in expected consumption and inflation.
  - Inflation defined as π_t = P_t / P_{t-1}. Under DE, cannot freely move P_t into the DE operator due to lagged reference dependence (see Bianchi et al. (2023), L’Huillier et al. (2023)).
- Housing FOC with DE:
  q_t / C^P_t = φ^P H^P_t + β E^θ_t[ q_{t+1} / C^P_{t+1} ]
  - q_t = Q_t / P_t; expected real house price accompanied by a surprise term.

### Entrepreneurs
- Production: Y^w_t = A_t (H^E_{t-1})^ν (L^E_t)^{1−ν}
  - Intermediate output sold to retailers at price P^w_t; X_t = P_t / P^w_t is the markup. A_t is TFP.
- Utility: U^E_t = log(C^E_t) with discount factor γ < β.
- Nominal budget constraint:
  P^w_t Y^w_t − W_t L^E_t + B^E_t + Q_t H^E_{t-1} ≤ P_t C^E_t + Q_t H^E_t + R_{t-1} B^E_{t-1}
- Collateral constraint (nominal asset holdings):
  B^E_t ≤ m E^θ_t[ q_{t+1} H^E_t R_t / π_{t+1} / π_t ]
  - m interpreted as loan-to-value (LTV) ratio.
- Euler and pricing conditions under DE:
  - Consumption Euler:
    1 / (π_t C^E_t) = γ E^θ_t[ 1 / C^E_{t+1} · R_t / π_{t+1} · 1 / π_t ] + λ_t q_t / C^E_t
  - Asset pricing for housing includes expected discounted marginal product of housing and continuation value plus effect on collateral constraint.
  - Borrowing constraint links house prices and interest rates to maximum borrowing; DE implies surprise terms in expected consumption, inflation, and real house price that amplify endogenous shock propagation.

- Financial accelerator vs deceleration:
  - Demand shock: fall in real house prices, output, and inflation tighten collateral constraint → reduces borrowing and activity → further lowers house prices (vicious cycle = financial accelerator).
  - Supply shock: output and prices move oppositely; spike in inflation reduces real debt stock and cushions shock (debt deflation channel can dampen shocks = financial deceleration).

### Final Goods and Retailers
- Retailers subject to Rotemberg (1982) adjustment cost, yielding log-linearized New Keynesian Phillips curve with DE:
  ˆπ_t = β E^θ_t[ ˆπ_{t+1} ] − (ε − 1)/ψ ˆx_t + ε^π_t
  - Hats denote deviations from steady state; ε^π_t is a cost-push shock.
  - Expected inflation under DE includes a surprise term, increasing endogenous variability.

### Central Bank
- Linearized interest rate rule:
  ˆR_t = ρ_R ˆR_{t-1} + (1 − ρ_R) [ ω_π ˆπ_t + ω_Y ˆY_t ] + ε^R_t
- Baseline rule may be modified to incorporate additional policy considerations (e.g., house price dynamics).

### Model Dynamics Under Diagnostic Expectations
- Log-linearized consumption Euler equation (patient households):
  E^θ_t[ ˆC^P_{t+1} ] − ˆC^P_t = E^θ_t[ ˆR_t − ˆπ_{t+1} − ˆπ_t ] − ˆπ_t
- DE operator decomposition:
  - E^θ_t[ ˆC^P_{t+1} ] = E_t[ ˆC^P_{t+1} ] + θ ( E_t[ ˆC^P_{t+1} ] − E_{t-1}[ ˆC^P_{t+1} ] ) with surprise ξ^C^P_{t+1} = E_t[ ˆC^P_{t+1} ] − E_{t-1}[ ˆC^P_{t+1} ].
  - E^θ_t[ ˆπ_{t+1} ] = E_t[ ˆπ_{t+1} ] + θ ξ^π_{t+1}, where ξ^π_{t+1} = E_t[ ˆπ_{t+1} ] − E_{t-1}[ ˆπ_{t+1} ].
  - Additional term θ (E_t[ ˆπ_t ] − E_{t-1}[ ˆπ_t ]) = θ ξ^π_t appears from log-linearization.
- Combined representation:
  E^θ_t[ ˆC^P_{t+1} ] − ˆC^P_t = E_t[ ∆ ˆC^P_{t+1} ] + ξ^C^P_{t+1}
  = E_t[ ˆRR_{t+1} ] + θ ξ^R_t − θ ξ^π_{t+1} − θ ξ^π_t
  - Where ˆRR_{t+1} = ˆR_t − ˆπ_{t+1}.
- Implications:
  - DE introduces surprise terms in both expected consumption growth and real interest rates (ξ^C^P, ξ^R, ξ^π_{t+1}, ξ^π_t), generating additional endogenous dynamics.
  - Marginal propensity to consume is higher under DE than RE: shocks boosting current income induce overly optimistic views of future output → overconsumption → subsequent disappointment and stronger contraction once reference expectations are revised.
  - Surprise in expected inflation and current inflation can lower the real interest rate (expansionary channel), interacting with consumption surprises to amplify consumption growth swings.
  - Financial accelerator amplifies these DE-driven swings, producing larger business cycle fluctuations.
  - For supply shocks, DE strengthens the debt deflation channel via surprises in expected and current inflation, which can reduce entrepreneurs’ debt burden and dampen shocks.

*Section 5, wpiea2024132-print-pdf.*

### 3.6    Model Parameterization

### 3.6    Model Parameterization

### Model calibration and key parameters
- Parameterization follows Iacoviello (2005) with exceptions noted in Table 1.
- Pricing uses Rotemberg (1982) rather than Calvo (1983), requiring:
  - Elasticity of substitution for intermediate goods: 11 (steady state markup of 10 percent).
  - Price adjustment parameter: 100.
- Two key parameters underpinning Diagnostic Expectations (DE):
  - J (time lag over which the RE revision is defined): benchmark J = 1.
  - Diagnosticity parameter θ: benchmark θ = 0.75 (in line with Bordalo et al. (2018) and L’Huillier et al. (2023)).
- Benchmark monetary policy rule parameters:
  - Weight on inflation ωπ = 2.
  - Weight on output gap ωY = 0.12.
  - Interest rate smoothing ρR = 0.73.
- Shock processes are calibrated and summarized at the end of Table 1.
- Sensitivity tests consider alternative parameterizations affecting the strength of DE, the source of exogenous variation, and monetary policy rules.

### Table 1 — Selected parameter values (as reported)
- Discount factors:
  - Patient household β 0.99
  - Entrepreneurs γ 0.98
- Weight on housing (services) φ 0.03
- Labor supply aversion η 1.01
- Housing share ν 0.1
- Elasticity of substitution ε 20
- Price adjustment cost ψ 140
- Loan-to-value ratio m 0.89
- Diagnosticity parameter θ 0.75
- Monetary policy rule:
  - Weight on inflation ωπ 1.5
  - Weight on output gap ωY 0.12
  - Interest rate smoothing ρR 0.73
- Shocks (standard deviation and persistence):
  - Monetary policy: σR 0.25; ρR 0.25
  - Cost push: σP 0.25; ρP 0.9
  - Total factor productivity: σA 0.25; ρA 0.9
  - Housing preference: σH 0.25; ρH 0.9

### Sensitivity and alternative calibrations
- Baseline DE calibration is conservative: θ = 0.75, J = 1.
- Alternative calibrations considered: J ∈ {1, 4} and θ ∈ {1, 2}.
- More distant memory (higher J) and θ > 1 produce more severe recessions.
- Example (reported comparison):
  - Under baseline: FA results in output decline of ́0.8 percent versus ́0.7 percent under baseline without FA (difference 0.1 percentage points).
  - DE (θ = 2, J = 4) results in output decline of ́2.5 percent (difference of 1.8 percentage points versus baseline).
  - Sum of individual contributions = ́1.9 percentage points; combined DE and FA yields an output decline of ́3.1 percent (difference versus baseline ́2.5 percentage points) — i.e., the whole is greater than the sum of parts.

### Main dynamic results and mechanisms
- Combining Diagnostic Expectations (DE) with the Financial Accelerator (FA) generates mutually reinforcing shock amplification.
- Demand shocks:
  - DE and FA jointly amplify demand shocks markedly through:
    - Asset price channel: declines in house prices tighten collateral constraints.
    - Debt deflation channel: lower inflation raises real debt service costs, depressing entrepreneurial consumption.
    - Overreaction under DE: agents become overly pessimistic after negative surprises, increasing the marginal propensity to consume and deepening recessions.
- Supply (cost-push) shocks:
  - FA tends to dampen supply shocks via the debt deflation channel because higher inflation loosens collateral constraints for entrepreneurs.
  - Under DE, cost-push shocks produce larger stagflationary outcomes (higher inflation and deeper recession); the debt deflation channel remains important and is strengthened by DE.
- Quantitative examples for a 1% (annualized) monetary policy shock:
  - Baseline (RE, no FA): immediate output decline of ́0.7 percent; cumulative output decline of ́2 percent after 30 quarters.
  - RE with FA: immediate output decline of ́0.8 percent; cumulative decline of ́3.5 percent after 30 quarters.
  - DE without FA: immediate output decline of ́1.2 percent (about twice the baseline immediate decline).
  - DE with FA: immediate output decline of ́1.3 percent (versus ́0.7 baseline).
- Key qualitative insight: DE strengthens the asset price channel and amplifies the surprises that feed into FA dynamics, producing larger declines in house prices and output — i.e., DE and FA are mutually reinforcing.

### Policy implications and trade-off analysis
- Diagnostic Expectations worsen the inflation-output volatility trade-off faced by policymakers.
- Policy frontiers:
  - Constructed from a quadratic loss L = Λ Var(π̂t) + (1 − Λ) Var(Ŷt) and solving for the optimal interest rate rule given the economy’s constraints.
  - DE with FA (DE FA) frontier lies further from the origin than baseline (RE, no FA), indicating a worse trade-off: at any given inflation volatility, DE FA has higher output volatility.
  - DE FA frontier is steeper: reducing inflation volatility requires progressively larger increases in output volatility under DE FA.
- Interaction with the debt deflation channel:
  - In some cases under DE, frontiers cross: beyond a certain level of inflation volatility, the FA model can be associated with lower output volatility due to a stronger debt deflation channel that more effectively damps supply shocks.
  - The intersection point occurs at lower inflation volatility under DE than under RE, reflecting DE’s enhancement of the debt deflation channel.

*Source: wpiea2024132-print-pdf — 3.6 Model Parameterization*

### 4.3    Simple monetary policy rules

### 4.3    Simple monetary policy rules

### Objective and policy rule setup
- Objective function assumed (following Adam and Woodford (2021)):
  W = E0 Σ_{t=0}^∞ β^t (ˆπ_t^2 + Λ_y ˆY_t^2 + Λ_h ˆπ_{t,h}^2)
  - Λ_y is the relative weight on output gap and Λ_h is relative weight on housing prices inflation.
- Generalized interest rate rule considered:
  ˆR_t = ρ_R ˆR_{t-1} + (1-ρ_R) (ω_π ˆπ_t + ω_Y ˆY_{t-1} + ω_q ˆq_t + ω_{∆Q} ˆ∆Q_t) + ε_{R,t}
  - Rule allows targeting level of house prices (ˆq_t) and growth rate of house prices (ˆ∆Q_t) in addition to standard inflation and output gaps.
- Four models analyzed: RE and DE, each with and without the Financial Accelerator (FA).
- Five policy rules evaluated:
  1. Flexible inflation targeting (FIT): ω_π ≥ 0, ω_Y ≥ 0, ω_q = ω_{∆Q} = 0.
  2. Strict inflation targeting (SIT): ω_π ≥ 0, ω_Y = ω_q = ω_{∆Q} = 0.
  3. Price-level targeting (PLT): central bank targets ˆP_t instead of ˆπ_t.
  4. Targeting level of house prices: ω_q ≥ 0 (with ω_π ≥ 0, ω_Y ≥ 0).
  5. Targeting house price growth: ω_{∆Q} ≥ 0 (with ω_π ≥ 0, ω_Y ≥ 0).
- Outcomes reported in Table 2: welfare cost (welfare in percent of forgone steady state consumption), policy coefficients (e.g., ω_Y), and standard deviations (σ, in percent) of inflation, output, house price level and house price growth.

### Main findings across models
- Baseline (RE without FA):
  - Optimal simple rule is FIT, which in this case coincides with SIT (Table 2, columns [1]-[2]).
  - PLT gives greater output stabilization (φ_Y ≠ 0) but slightly higher inflation volatility, increasing welfare losses relative to FIT.
- Introduction of the Financial Accelerator (FA):
  - Welfare losses increase across the board relative to the baseline.
  - FIT outperforms SIT and PLT when output is taken into account.
  - Targeting the level of house prices offers no merit: under column [7] the weight on the level of house price is zero (φ_q = 0), making columns [4] and [7] equivalent.
  - Targeting the growth rate of house prices yields the lowest welfare losses (Table 2, column [8]):
    - Taming the asset price channel by targeting house price growth leads to a sizable reduction in output volatility with only a slight increase in inflation volatility.
    - Advantage: policy maker need not infer the underlying value of the asset; targeting growth avoids judgment on asset price misalignment and is more robust to information asymmetries.
- Deterministic Expectations (DE) variants:
  - DE without FA: welfare losses associated with policy rules increase relative to RE; PLT yields the most favorable welfare outcome (Table 2, columns [9]-[11]).
    - Path dependence of PLT amplifies debt deflation channel and, combined with stronger sensitivity to output gap, lowers business cycle fluctuations at the cost of modestly higher inflation volatility.
  - DE with FA (Table 2, columns [12]-[16]):
    - No value in targeting the level of house prices (overlap between column [15] and FIT under column [12]).
    - PLT is superior to FIT and SIT.
    - Policy rule targeting growth of house prices performs best by welfare score (column [16]):
      - The rule responds aggressively to fluctuations in house price growth (φ_{∆Q} = 2.18), suppressing the asset price channel (dampening demand shocks) while accommodating some inflation volatility to harness debt deflation benefits (strengthened under DE and helpful against supply shocks).

### Interpretation and policy implications
- Consistent result: no case for targeting the level of house prices within this framework.
- There is merit in considering the growth of asset prices when formulating monetary policy:
  - Targeting house price growth lowers welfare costs and improves stabilization properties by reducing output volatility more than it raises inflation volatility.
  - Targeting growth is robust to information frictions because it avoids inferring asset fundamental values.
- FIT (including response to output) generally outperforms strict inflation-only strategies when FA is present.

### Sensitivity exercises (summary)
- Robustness checks summarized in Appendices D and E yield main takeaways:
  - Benefits of targeting house price growth become more pronounced when amplification from DE is enhanced (including under J = 4, and when theta = 2).
  - Introducing indexed debt (to dampen the debt deflation channel) and recomputing optimal simple rules:
    - Indexed debt can negate wealth effects and impact on the collateral constraint associated with inflation but does not remove current inflation surprises associated with DE; it has a more limited effect and does not alter the main conclusions.
  - Different shock combinations (e.g., only supply shocks or only cost-push shock) do not overturn the main findings.
  - Lower loan-to-value ratio of 50 percent (on its own and with other sensitivity tests; see bottom row of Table 3):
    - Targeting house price growth still yields the best welfare outcomes.

### Concluding summary
- The financial accelerator model with Diagnostic Expectations (DE) generates mutually reinforcing shock amplification, particularly for demand shocks; supply shocks can be attenuated via a strengthened debt deflation channel under DE.
- The inflation-output volatility trade-off worsens under DE.
- Across model variants and robustness checks, targeting house price growth consistently reduces welfare costs and improves stabilization properties, whereas targeting the level of house prices shows no value.
- Two avenues for future research highlighted:
  - Assess whether the case for leaning against the wind by targeting house price growth holds when macroprudential tools exist and can be effectively implemented by a relevant authority.
  - Empirically evaluate a model combining DE, financial frictions, and other empirically-relevant rigidities.

*Source: wpiea2024132-print-pdf - 4.3    Simple monetary policy rules*

### References

### References

### Model equations (Appendix A)
- The appendix tabulates the full set of equilibrium conditions for the financial accelerator model with Diagnostic expectations. Key structural notes:
  - In the linearized model, θ=0 results in the Rational Expectations model.
  - As in Iacoviello (2005), setting b_Et = B_ER shuts off the asset price channel (and therefore the financial accelerator mechanism).
  - The model with indexed debt implies that π_t+1 terms in the consumption Euler equations drop out; for example, equation (A2) becomes:
    - 1/π_t 1/C_Pt = β E_θt [ R_t C_Pt+1 1/π_t ] (as stated in the text).
  - The analysis considers eight models that are nested in the system of equations presented above (DE or RE, with and without FA, nominal or indexed debt).

### Housing preference and TFP shocks (Appendix B)
- Housing preference shock (Figure 8):
  - Simulated by decreasing φ_Pt which alters the marginal rate of substitution between housing and consumption.
  - Immediate impact: sharp house price decline.
  - Mechanism: decreased housing demand tightens the collateral constraint, which initially dominates and depresses entrepreneurial activity, acting as a drag on the economy.
  - Recovery: as house prices recover, the collateral constraint loosens and entrepreneurial activity recovers.
  - Comparative dynamics: shock effects are more prevalent amid the FA models.
  - Interaction with debt-deflation: because output and inflation move in opposite directions (as in a supply shock), the debt deflation channel helps dampen the impact of the shock, especially under DE.
- TFP shock (Figure 9):
  - A temporary decline in TFP behaves similarly to an adverse supply (cost-push) shock in the FA model.
  - Effects include decreased housing demand by patient households as consumption decreases, leading to lower house prices and a tighter collateral constraint (partially offset by higher inflation).
  - Main takeaway: DE results in larger macroeconomic responses relative to RE in this scenario.

### OSR impulse response analysis (Appendix C)
- Setup:
  - Analysis focuses on two main shocks: a monetary policy shock and a cost-push shock.
  - Displays four macroeconomic aggregates: output, the inflation rate, the policy rate, and the level of real house prices.
  - Models compared:
    1. Baseline RE model without the FA.
    2. RE model with the FA.
    3. DE model with the FA.
  - Considered optimized simple rules: FIT and PLT in the baseline model (SIT overlaps with FIT or is inferior in welfare), and rules that target the growth rate of house prices in models with the FA (the rule including the level of house prices overlapped with FIT, i.e., φ_q = 0).
- Monetary policy shock (Figure 10; discussed with Figure 12 references):
  - Baseline model:
    - FIT is associated with lower output and inflation fluctuations relative to PLT, consistent with lower welfare cost for FIT.
  - With FA:
    - The rule targeting house price growth prescribes less aggressive policy tightening.
    - Less severe decline in house prices attenuates the asset price channel, stabilizes output, and yields a milder recession relative to FIT and PLT cases.
  - DE with FA:
    - Starker differences: targeting house price growth can produce an endogenous initial decline in the nominal interest rate, leading to less violent business cycle gyrations.
- Cost-push shock (Figure 11; discussed with Figure 13 references):
  - Baseline model:
    - PLT is associated with higher inflation volatility (owing to the historically-dependent nature of the policy regime) and higher welfare costs relative to FIT.
  - With FA:
    - The rule targeting house price growth prescribes an on-impact monetary loosening, resembling optimal policy responses discussed elsewhere.
    - Tempering the asset price channel confers notable stabilization benefits and yields a less severe initial contraction.
    - Through the debt deflation channel, a slightly higher increase in the inflation rate helps soften the shock’s impact.
  - DE with FA:
    - Monetary loosening occurs across optimized rules; the initial policy rate cut is most pronounced under the rule targeting house price growth.
    - The shallower decline in house prices mitigates the asset price channel and yields a shallower recession.
  - Summary conclusion:
    - Rules that target the growth rate of house prices attenuate the asset price channel and, even without directly targeting output, help reduce the amplitude of business cycle fluctuations.

### OSR sensitivity and robustness (Appendices D and E)
- The appendices present robustness checks and sensitivity analysis for the OSR results (labeled "OSR: Robustness checks" and "OSR: Robustness checks" respectively).
- These sections are intended to validate the stability of the optimal simple rule conclusions across alternative specifications and parameter perturbations (detailed figures and numerical outcomes are presented in the appendices).

*The Diagnostic Financial Accelerator — Working Paper No. WP/2024/132*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024132-print-pdf.pdf_